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Parikh vectors
7PCD_1 5PUN_1 5RBZ_1 Letter Amino acid
19 11 37 A Alanine
23 14 3 R Arginine
22 13 8 E Glutamic acid
5 10 2 H Histidine
11 7 2 M Methionine
13 5 51 T Threonine
9 4 12 N Asparagine
6 1 2 C Cysteine
11 8 12 Q Glutamine
19 5 20 I Isoleucine
8 6 22 F Phenylalanine
23 13 22 D Aspartic acid
23 7 46 G Glycine
16 6 19 K Lycine
18 7 58 S Serine
6 0 6 W Tryptophan
28 10 31 V Valine
37 18 27 L Leucine
19 6 22 P Proline
11 5 17 Y Tyrosine 

7PCD_1|Chain A|Receptor tyrosine-protein kinase erbB-2|Homo sapiens (9606)
>5PUN_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
>5RBZ_1|Chain A|Endothiapepsin|Cryphonectria parasitica (5116)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
7PCD , Knot 141 327 0.83 40 194 313 
SGAAPNQALLRILKETELRKVKVLGSGAFGTVYKGIWIPDGENVKIPVAIKVLRENTSPKANKEILDEAYVMAGVGSPYVSRLLGICLTSTVQLVTQLMPYGCLLDHVRENRGRLGSQDLLNWCMQIAKGMSYLEDVRLVHRDLAARNVLVKSPNHVKITDFGLARLLDIDETEYHADGGKVPIKWMALESILRRRFTHQSDVWSYGVTVWELMTFGAKPYDGIPAREIPDLLEKGERLPQPPICTIDVYMIMVKCWMIDSECRPRFRELVSEFSRMARDPQRFVVIQNEDLGPASPLDSTFYRSLLEDDDMGDLVDAEEYLVPQQG
5PUN , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA
5RBZ , Knot 170 419 0.81 40 190 380 
MSSPLKNALVTAMLAGGALSSPTKQHVGIPVNASPEVGPGKYSFKQVRNPNYKFNGPLSVKKTYLKYGVPIPAWLEDAVQNSTSGLAERSTGSATTTPIDSLDDAYITPVQIGTPAQTLNLDFDTGSSDLWVFSSETTASEVDGQTIYTPSKSTTAKLLSGATWSISYGDGSSSSGDVYTDTVSVGGLTVTGQAVESAKKVSSSFTEDSTIDGLLGLAFSTLNTVSPTQQKTFFDNAKASLDSPVFTADLGYHAPGTYNFGFIDTTAYTGSITYTAVSTKQGFWEWTSTGYAVGSGTFKSTSIDGIADTGTTLLYLPATVVSAYWAQVSGAKSSSSVGGYVFPCSATLPSFTFGVGSARIVIPGDYIDFGPISTGSSSCFGGIQSSAGIGINIFGDVALKAAFVVFNGATTPTLGFASK

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(7PCD_1)}(2) \setminus P_{f(5PUN_1)}(2)|=129\), \(|P_{f(5PUN_1)}(2) \setminus P_{f(7PCD_1)}(2)|=55\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:011110011101100001001011101111010011111010010111110110000010100011001011111101010011110100010110011101011001000010110001101010110110010010110001110011100100101001111011010000001011011101111001100010000011001101101101110100111100110110010011011100101011110011100000101001100100110010011110000111101100010001100001101101000111001
Pair \(Z_2\) Length of longest common subsequence
7PCD_1,5PUN_1 184 3
7PCD_1,5RBZ_1 172 4
5PUN_1,5RBZ_1 182 3

Newick tree

 
[
	5PUN_1:93.26,
	[
		7PCD_1:86,5RBZ_1:86
	]:7.26
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{483 }{\log_{20} 483}-\frac{156}{\log_{20}156})=96.2\)
Status Protein1 Protein2 d d1/2
Query variables 7PCD_1 5PUN_1 121 87.5
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]